Europe (and the "West" aka where Europe sent its surplus citizens) went through its fertility transition [0] earlier than the rest of the world. Hence it increased its relative size as it experienced a high fertility / low death period, and decreased its relative size after it went to a low fertility / low death while part of the rest of the world went into their high fertility / low death phase.
So we can probably expect currently high population growth to transition to the low fertility / low death endgame. China is already well past it.
As far as I know, no country went back to a high fertility / low death scenario, though countries stabilized at various level of low fertility.
Most Western countries have institutions that rely on population growth (high debt that would lead to enormous debt per capita if the number of people decreases, unmaintained infrastructure, pay-as-you-go pensions...). So they will either have to fix their fertility rate in a dramatic way, massively reform many of the core institutions of society, or have high immigration & handle it well.